Number Properties

If x = 989 and y = 991, what is the remainder of xy/9? How would you setup this problem?

(990 - 1)*(990+1) = (990^2 - 1^2) 990^2 is divisible by 9 and 9/9 - 1/9 = 8/9 so the remainder is 8

What do perfect squares always end in?

0, 1, 4, 9, 6, 5

What is the remainder for xyz/2 ? 1) 6xy = even 2) 9zx = even

1) 6xy = even xy could be even or odd - NS 2) 9zx = even z or x is even, so the remainder is 0 - S

Does A have more unique prime factors than B? 1) A = B^2 2) A > B

1) A = B^2 A and B will have the same # of unique prime factors 2) A > B NS

If 1000! / 5^n is an integer, what is the largest possible value of n?

1000/5^1 = 200 1000/5^2 = 40 1000/5^3 = 8 1000/5^4 = 1 1000/5^5 = 0 200 + 40 + 8 +1 n = 249 *when multiple primes divide by prime with the least amount in the factorial

How do you solve 45x = 0.4545?

45x = .4545 x = (.4545/45) = [45(0.0101)]/45 = 0.0101

When a certain number x is divided by 82, the remainder is 5. What is the remainder when x + 11 is divided by 41?

82(x/82 = Q + 5/82) x = 82Q + 5 x + 11 = 82Q + 5 + 11 = 82Q + 16 (82Q + 16)/41 = 2Q + 16/41 16 is the remainder

When does a decimal terminate?

A decimal will terminate only if the denominator of the reduced fraction has a prime factorization that contains only 2's or 5's or both. If the prime factorization contains anything other than 2's or 5's, the decimal will not terminate.

Are 0 and 1 prime numbers?

A prime number is divisible by 1 and itself. 0 is not prime because it cannot be divisible by itself. 1 is not a prime number because it is only divisible by itself and not divisible by 1 and another number.

What does 5 x 2 do?

Always creates trailing zero

The integer Z is divisible by 15 and 20. All of the following must be a factor of Z except for which of the following?

Find the LCM (least common multiple) 15: 3, 5 20: 2, 2, 5 = 2^2, 5 LCM: 2^2 * 3 * 5 = 60 Z at minimum divisible by LCM: 2*3*5 = 30 Z at maximum divisible by 2^2 * 3 * 5 = 60

If 72 factors to 2^3 and 3^2 and 120 factors to 2^3, 3^1, and 5^1 What is the greatest common factor and least common multiple?

GCF = smallest of numbers in common 2^3 * 3^1 LCM = largest of each 2^3 * 3^2 * 5^1

Does A have more unique prime factors than B if A = B^2?

If A = B^2, it must share the exact same unique prime factors. Squaring an integer does not change the unique prime factors are that are in that integer. This is as good as saying A = B. (this would be sufficient to answer the question)

When will the decimal of a fraction terminate?

If the denominator of the reduced fraction has a prime factorization that contains only 2's or 5's

How do you find the Greatest Common Factor?

Pick the smallest prime #'s

If a, b, c, and d are integers; w, x, y, and z are prime numbers; w<x<y<z and (w^a)(x^b)(y^c)(z^d) = 660, how would you find a, b, c, and d?

Prime factor 660, so you get (2^2)(3^1)(5^1)(11^1) a = 2 b = 1 c = 1 d = 1

A bag of n peanuts can be divided into 9 smaller bags with 6 peanuts left over. Another bag of m peanuts can be divided into 12 smaller bags with 4 peanuts left over. Which of the following is the remainder when nm is divided by 18?

Setup equations: n/Q = 9 + 6/Q => n = 9Q + 6 m/Z = 12 + 4/Z => m = 12Z + 4 (9Q + 6)*(12Z + 4) = (108QZ + 36Q + 72Z + 24)/18 All are divisible by 18, 24 is not, which leaves a remainder of 6

If x, y, and z are positive integers such that x^2 + y^2 + z^2 = 64,470, which of the following could be the values of x, y, and z? (How would you solve?) 1) x = 115, y = 146, z = 173 2) x = 114, y = 142, z = 171

Square the units digit and add the units digits of the squared number to confirm whether or not it is equal to 0

How do you find the Least Common Multiple?

The largest prime #'s

What are the divisibility rules for 8?

The number must be even and the last three digits must be divisible by 8. The number is also divisible by 8 if the original number ends in three zeros (000).

What is the largest number that must be a factor of the product of any four consecutive positive integers?

The product of any set of n consecutive positive integers is divisible by n! 4! = 4 x 3 x 2 x 1 = 24

If x is odd, what is the remainder when x is divided by 2?

The remainder when any odd number is divided by 2 will always be 1.

What is the remainder formula?

x/y = Q + r/y

If some number x has y unique prime factors, then x^n will have how many unique prime factors?

x^n will has have y unique prime factors

Determine the largest number of a prime number x that divides into y!

y/x^x until 0 and add

Is y divisible by 3 if y^2 is divisible by 9? (Think about the math)

y^2 / 9 = k y^2 = 9*k *take sqrt of both sides* y = 3 sqrt(k) y must be divisible by 3